In Introduction to Fixed Income Analytics by Frank Fabozzi, p. 41, there is an example how to calculate the theoretical spot rate of a 1.5 year treasury bond with a 3.5% annual interest and semiannual coupons.

\begin{array} {|c|c|c|c|c|} \hline \text{Period} & \text{Years} & \substack{\text{Annual Yield to}\\\text{Maturity (BEY) (%)}} & \text{Price} & \substack{\text{Spot Rate}\\\text{(BEY) (%)}}\\ \hline 1 & 0.5 & 3.00 &-& 3.0000 \\ 2 & 1.0 & 3.30 &-& 3.3000 \\ 3 & 1.5 & 3.50 &100.00& \text{?}\\ \hline \end{array}

The first two rows are the zero-coupon treasury bills.

The cash flow of the 1.5 year treasury bond is obviously:

0.5 year: 0.035 × \$100 × 0.5 = \$1.75
1.0 year: 0.035 × \$100 × 0.5 = \$1.75
1.5 year: 0.035 × \$100 × 0.5 + 100 = \$101.75

He now claims that the present value of the cash flows is:

$$\mathrm{PV}(z_1, z_2, z_3) = \frac{1.75}{(1+z_1)^1} + \frac{1.75}{(1+z_2)^2} + \frac{101.75}{(1+z_3)^3}$$ where

$z_1 =$ one-half the annualized 6-month theoretical spot rate
$z_2 =$ one-half the annualized 1-year theoretical spot rate
$z_3 =$ one-half the annualized 1.5-year theoretical spot rate.

If we solve $\mathrm{PV}(3.00, 3.30, z_3) = 100$ for $z_3$, we are supposed to get the theoretical spot rate for the 1.5 year treasury bond as described above.

But why can he do the whole example on semiannual intervals? The 1-year treasury bill does not pay any semiannual coupons, right? Where does that compounding come from?

Why isn't it: $$\mathrm{PV}(z_1, z_2, z_3) = \frac{1.75}{1+z_1} + \frac{1.75}{2\cdot z_2} + \frac{101.75}{(1+z_3)^3}\; ?$$

OK, BEY is given as effective semiannual rate. The convention is simply defined that way. So the question has been resolved.


1 Answer 1


The question really is what is the discount factor for a payment in one year assuming semiannual compounding? Because then your present value is simply 1.75 times this discount factor.

If you have $k$ periods of compounding, a payment of \$1 in $n$ years worth today \begin{align*} \frac{1}{\left(1+\frac{r}{k}\right)^{n\cdot k}}, \end{align*} where $r$ is the annualised spot rate. In your case, $k=2$ for semiannual compounding and $n$ is firstly $0.5$, then $1$ and finally $1.5$. This gives rise to the following three discount factors \begin{align*} \frac{1}{\left(1+\frac{r_{0.5}}{2}\right)^{\frac{1}{2}\cdot 2}} &=\frac{1}{1+\frac{r_{0.5}}{2}}, \\ \frac{1}{\left(1+\frac{r_{1}}{2}\right)^{1\cdot 2}} &=\frac{1}{\left(1+\frac{r_{1}}{2}\right)^{2}},\\ \frac{1}{\left(1+\frac{r_{1.5}}{2}\right)^{\frac{3}{2}\cdot 2}} &=\frac{1}{\left(1+\frac{r_{1.5}}{2}\right)^{3}}, \end{align*}

Regarding the logic behind it: Your bond has three payments and you need to find their value today. So, you decompose the coupon-paying bond into different zero-coupon bonds (ZCB) with a face value of \$1 and you see that \begin{align*} T-Bond & = 1.75 \cdot ZCB(0.5y) + 1.75\cdot ZCB(1y) + 101.75 \cdot ZCB(1.5y) \\ &= 1.75\cdot \frac{1}{\left(1+\frac{r_{0.5y}}{k}\right)^{\frac{1}{2}\cdot k}} + 1.75\cdot \frac{1}{\left(1+\frac{r_{1y}}{k}\right)^{1\cdot k}} + 101.75 \cdot \frac{1}{\left(1+\frac{r_{1.5y}}{k}\right)^{\frac{3}{2}\cdot k}} \end{align*} So, you use the 1y spot rate because you got a payment in 1y and need to discount it back to today. What compounding you use, does not matter, so one can write the above equation for instance in terms of $e^{-r T}$ and still get the same result.


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